ON THE INFIMUM OF THE HAUSDORFF METRIC TOPOLOGIES

Given a metrizable space X and a compatible metric d, one defines the Hausdorff metric topology H-d and the upper and lower Hausdorff topologies corresponding to d, H-d(+) and H-d(-) respectively, on the collection Ce(X) of all closed subsets of X. In this paper we consider the infima tau, tau(+) and tau(-), of the topologies H-d H-d(+) and H-d(-) respectively, where d runs over the set M(X) of all compatible metrics on X. These topologies are sequential, that is, they are completely characterized by convergent sequences. In particular, the topologies tau(+) and tau(-) are investigated in detail: a suitable topology U+ is defined which has the same convergent sequences as tau(+), and the lower Vietoris topology V- plays a similar role with respect to tau(-). We show that, in general, the equality tau = tau(+) V tau(-) does not hold. We also show that tau is a tau(2)-topology on C(X) if and only if X is locally compact.